It seems that the function $\sin x$ is both half discrete-analytic and weak discrete-analytic. But the function $\sin \frac{\pi x}2$ is only weak discrete-analytic.

So what is the maximum $a$ such that $\sin ax$ is half discrete-analytic?

I also interested to know at which $a$ $\sin ax$ is weak discrete-analytic. For example, is $\sin 3x$ weak discrete-analytic or not?

This question is motivated by my old search for a natural fractional integration and integration constant (integral analog of Ramanjuan sum). I want to find a natural generalization of Newton series to functions whose Newton series normally diverges. Once that accomplished it would be possible to find Newton expansions for consecutive derivatives of a function and by analytically continuing them into negative domain, get natural integral. Unfortunately finding natural fractional integral of even such simple function as $\sin x$ requires building Newton expansion for $\sin \frac{\pi x}2$ which diverges.

Closed form here involves some ${}_2F_1$ functions, so I cannot provide proofs. Numerically, though, it seems that $\sin(ax)$ is weak discrete-analytic for $a$ up to some value just above $3$. Maybe $\pi$, I guess.

In this diagram we have $\sin(ax)$ in blue (for $x=5.678$); and 5000 terms for the Newton series in red. They agree for $a$ up to about $3.1$ or $3.2$...